Analytical study of temperature dependent RKKY interaction

Abstract

We develope a new method to calculate the temperature dependent behavior of ideal Fermi gas system using complex theory and Eluer-Maclaurin integration formular. Our method does not require any restriction on the properties of integrand, that is to say, integrand is nonsingular and not too rapidly varying in the neighborhood of $\epsilon = \mu$, as Sommerfeld expansion method does. Using this method we investigate the spin susceptibility of free electron gas in one and three dimension at arbitrary temperature and derive analytical expression of F(q) and $\chi$(r). Our analytical studies show that the amplitude decay of three dimensional $\chi_3$(r), which is algebraic at T = O, becomes an exponential form in the long range limit as a function of both temperature and distance contrary to Darby``s gaussian prediction and the oscillation period varies with temperature. We compare our analytical results with numerical simulation obtained by using FFT(Fast Fourier Transform) method. Our numerical studies show that the analytical results are accurate enough if we consider to second order correction term both in the short and long range region. On the other hand, the Sommerfeld expansion method does not produce correct long range behavior even though short range behavior is fairly accurate.